Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gromov-Hausdorff limits of immortal Kähler-Ricci flows

Man-Chun Lee, Valentino Tosatti, Junsheng Zhang

Abstract

We show that the normalized Kähler-Ricci flow on a compact Kähler manifold with semiample canonical bundle converges in the Gromov-Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model, as conjectured by Song-Tian's analytic mimimal model program.

Gromov-Hausdorff limits of immortal Kähler-Ricci flows

Abstract

We show that the normalized Kähler-Ricci flow on a compact Kähler manifold with semiample canonical bundle converges in the Gromov-Hausdorff topology to the metric completion of the twisted Kähler-Einstein metric on the canonical model, as conjectured by Song-Tian's analytic mimimal model program.
Paper Structure (16 sections, 18 theorems, 180 equations)

This paper contains 16 sections, 18 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The twisted Kähler-Einstein metric $\omega_{\rm can}$
  4. Known results about $\omega(t)$
  5. Known results about $\omega_{\rm can}$
  6. Hölder bound on the potential and distance function of $\omega_{\rm can}$
  7. Hölder bound for the potential of $\omega_{\rm can}$
  8. Hölder bound for $d_{\rm can}$
  9. Reduction to the key estimate
  10. Reduction to an upper of $d_{\mathop{\mathrm{can}}\nolimits}$ by $d_t$
  11. Further reduction to an upper bound of $d_{\mathop{\mathrm{can}}\nolimits}$ by the $\mathcal{L}$-distance
  12. Proof of the main theorem
  13. An almost-avoidance principle
  14. Integrability of $\omega_{\mathop{\mathrm{can}}\nolimits}$ on the almost regular set
  15. The choice of the regions
  16. ...and 1 more sections

Key Result

Theorem 1.2

In the setting of Conjecture con, we have that as $t\to\infty$ the flow $(X,\omega(t))$ converges in the Gromov-Hausdorff topology to $(Y,d_{\mathrm{can}})$. In particular, Conjecture con holds.

Theorems & Definitions (34)

  • Conjecture 1.1
  • Theorem 1.2
  • Theorem 2.1
  • Remark 2.2
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • Lemma 3.4
  • proof
  • ...and 24 more